I'm looking for a special effect on vocals, which is generally used to emphasize the introduction of vocals and sounds like a swallowing of the first syllable, or like a reverse reverb. It's hard to explain, so here's an example between 0:11 and 0:15 :

Duplicate the vocal track, and then reverse it. Put a reverb plugin on it. Then bounce that down to a new track, and reverse it again. You'll then need to cut it to only have the little part that has the effect you want.